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Rigidification of higher categorical structures

Giovanni Caviglia and Geoffroy Horel

Algebraic & Geometric Topology 16 (2016) 3533–3562

Given a limit sketch in which the cones have a finite connected base, we show that a model structure of “up to homotopy” models for this limit sketch in a suitable model category can be transferred to a Quillen-equivalent model structure on the category of strict models. As a corollary of our general result, we obtain a rigidification theorem which asserts in particular that any Θn–space in the sense of Rezk is levelwise equivalent to one that satisfies the Segal conditions on the nose. There are similar results for dendroidal spaces and n–fold Segal spaces.

internal operads, internal n-categories, limit sketches, model categories
Mathematical Subject Classification 2010
Primary: 18C30, 18D35, 55U35
Secondary: 18D05, 18D50
Received: 13 November 2015
Revised: 16 May 2016
Accepted: 16 May 2016
Published: 15 December 2016
Giovanni Caviglia
Institute for Mathematics, Astrophysics, and Particle Physics
Radboud University Nijmegen
Heyendaalseweg 135
6525 AJ Nijmegen
Geoffroy Horel
Max Planck Institute for Mathematics
Vivatsgasse 7
D-53111 Bonn